Derive Perpetual Bond Price

It is known that a perpetual bond with coupon $c$ has price $$P=\frac{c}{r}$$ How do you get to this price? Is $r$ stated in discrete or continuous compounding?

A Consol Bond is a bond that pays an annual coupon of c every year. Therefore its price is $P=\frac{c}{1+r}+\frac{c}{(1+r)^2}+\cdots$. Factoring out the c and using the known formula for a geometric series, namely $u+u^2+u^3+\cdots = \frac{u}{1-u}$ we get $P=c[\frac{1}{1+r}/(1-\frac{1}{1+r})]=\frac{c}{r}$

Clearly this is a discrete compounding, not continous compounding formula.

• Thank you. What would be the price in continuous compounding? Dec 13 '15 at 0:28
• Assuming one coupon payment per year, by a very similar reasoning it is $P=c\frac{e^{-r}}{1-e^{-r}}$ Dec 13 '15 at 0:58
• In this link it finds that $r_c=\ln r-1$. But this would contradict your formula for $P$ in continuous compounding? Dec 13 '15 at 1:48
• The correct relationship is $r_c=\ln (1+r)$ (see for example post of May 1, 2013 in your link, with n=1) Dec 13 '15 at 2:17
• Thanks, then we get $r=e^{r_c}-1\Rightarrow P=\frac{c}{r}=\frac{c}{e^{r_c}-1}$. This is the same result as in your formula above when extending by $\frac{e^r}{e^r}$. Dec 13 '15 at 2:25

By definition; to get your required annual perpetual return of r, you trivially pay 1 USD up-front to get r USD annually. To get those annual payments from the consol bond in question you need to have r/c bonds (each paying c USD annually). To get those bonds for your 1 USD up-front payment, they have to sell at the price of c/r USD which is hereby demonstrated.

• Very nice intuitive proof. May 15 '19 at 15:46